COMPARISON OF SOME WIDELY PROPOSED METHODS BY CRITERIA

for electing one person

by Mike Ossipoff

To write to Mike write nkklrp before the "@" sign, and then write hotmail.com after the "@" sign.

Comparisons will be in terms of criteria & tests that I've previously defined.

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Definitions of some widely proposed methods:

I've already defined Plurality, Approval & Condorcet's method (in two versions). I'd like to define 5 more methods that are sometimes proposed, in order of how widely proposed they are. First I'll list them:

1. Instant Runoff:

Instant Runoff, as it's called by its U.S. promoters, can be defined by the following brief rule:

Voters rank candidates in order of preference.

Repeatedly, eliminate from the rankings the candidate who is currently at the top of fewest rankings.

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As usually defined, Instant Runoff includes a stopping rule: If a candidate aquires a majority, then the count stops & he/she wins. But that stopping rule doesn't affect the outcome, which is why it's left out of the brief definition. The stopping rule probably is intended to reduce count labor.

Usually Instant Runoff ("IRV") is defined more wordily by its promoters. They usually speak of a vote that one gives to one's favorite. The candidate with fewest votes is eliminated, and if the candidate who has your vote is eliminated, then your vote goes to your next choice. This elimination & transfer continues till someone has a majority of the votes, or till there's only 1 candidate left.

I state this unnecessarily long definition because that's how IRV will usually be found.

2. Borda:

De Borda was a contemporary of Condorcet. They both proposed their respective rank-balloting count rules at that time, and, regrettably, Borda's rule is the one that has become more well known, and has received some use. That's probably because Condorcet's method required a quite unfeasible amount of count labor for public elections and other big elections with many alternatives, before computers were available.

Borda, as usually proposed, requires
Borda count rule: With N candidates, your 1st ranked candidate receives N-1 points from you. Your 2nd ranked candidate receives N-2 points from you. And so on. Your lowest-ranked candidate receives zero points from you. So each rank receives one less point from you than the rank above it.

Borda is used to elect the "Most Valuable Player" in either U.S. football or U.S. baseball. It's an election in which sportscasters vote.

Of course Borda wouldn't be so bad if voters could choose how many candidates to rank. And, to add more freedom to the method, why shouldn't a voter be able to give any point score to a candidate, instead of a rigid decreasing sequence?

If voters may give any point score to a candidate, that's been called a flexible point system, among other names. Of course there would be a maximum point rating allowed for a candidate, and the minimum would typically be zero, though some suggestions include negative ratings too--it's really the same method either way.

Strategy in such a method, for the voter who wants to maximize his/her utility expectation, would be to give maximum points to every candidate for whom he/she would vote in Approval, and minimum points to every candidate for whom he/she wouldn't vote in Approval.

More about strategies for various simple methods can be found in Making Multicandidate Elections More Democratic by Samuel Merrill.

So strategically, a flexible point system is equivalent to Approval. So there'd be no need to bother with the balloting requirements for flexible points, since it just amounts to Approval anyway.

3. Olympic 0-10:

I've only heard this proposal once, in a letter to the editor, but it's worth mentioning, because rating things from 0 to 10, or from 1 to 10, is such a familiar thing. Even though 0-10, like any flexible point system, is equivalent to Approval, as described above, 1-10 might still be worth proposing it the familiarity of that kind of ratings is considered a decisive factor in its acceptability.

4. Copeland

Copeland is a pairwise count method, but it fails the criteria that I defined in the Strategy Criteria article.

Copeland's count rule:

A candidate's Copeland score is calculated by subtracting the number of candidates who beat him from the number of candidates whom he beats (where A beats B if more voters rank A over B than vice-versa). The winner is the candidate with the highest Copeland Score.

5. Dodgson's method

Dodgson was the author who used the pseudonym "Lewis Carroll". He suggested evaluating count rules by whether they pick the winner of the following method pairwise-count rule:

Elect the BeatsAll winner if one exists. Otherwise, pick the candidate who could be made into a BeatsAll winner by reversing as few preferences as possible.

Some people substitute "ignoring" for "reversing", but it's so similar that it's reasonable to still call that Dodgson's method.

Dodgson sounds appealing, but Dodgson fails the criteria that I defined in the Strategy Criteria article.

Some criterion compliances & noncompliances of these methods:

In the Approval & Condorcet articles I've already stated some criteria that they meet, and some tests that they pass. Those won't be repeated here, to save space.

Now I'll list the methods one by one, saying something about their compliances & noncompliances.

Condorcet:

SD & SSD meet all of the criteria that I've defined in these articles, except for FBC & SARC, which are met by no method except for Approval.

PC meets the same criteria, except that it doesn't meet GSFC, and doesn't strictly meet SDSC or SrDSC (though it meets them for all practical purposes). Among the criteria in the Traditional Criteria article, PC meets the Majority Criterion & the Condorcet Criterion & Monotonicity, but it fails Condorcet Loser, Majority Loser, & Mutual Majority.

But, for one thing, some of those criteria are bottom-end criteria, failed by PC only when every candidate is beaten, and when there's no really obvious right choice anyway. And it would be a peculiarly popular Condorcet loser, for example, who would have fewer people ranking someone over him than any other candidate has. And the Mutual Majority Criterion is really just about how well methods avoid a special particular kind of split-vote problem, and doesn't have the generality of WDSC and SFC. To me, then, WDSC, and especially SFC are much more important than the criteria that PC fails.

I re-emphasize that SD & SSD meet all of the criteria defined in these articles except for FBC & SARC, which are met by no method except for Approval.

I should add that PC can be made to meet the academic criteria listed in this article, in the following way:

Use PC to choose from the Smith set.

The Smith set is the smallest set of candidates such that every candidate in the set beats every candidate outside the set.

The resulting method has been called Smith//Condorcet, or SC. Though SC meets all the academic criteria listed in this article, it still falls short of SD & SSD, because they meet GSFC, and strictly meet SDSC & SrDSC. Also, SD & SSD are considered to be the best interpretations of what Condorcet meant in one of his proposals. And they have a natural & obvious interpretation & justification.

Approval:

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The criteria in the Traditional Criteria article, except for Monotonicity, are intended only to apply to rank-methods. If they were applied to Approval, it would pass them.

Approval passes Monotonicity, and is the only method that meets FBC & SARC. It meets WDSC & UUCC.

IRV:

IRV fails all of the criteria listed in the Strategy Criteria article. In other words, it fails all of the criteria that I consider important, the ones that relate to the lesser-of-2-evils problem, and all but the weakest that relate meaningfully to majority rule. Specifically, it fails FBC, SARC, WDSC, SDSC, SrDSC, SFC, GSFC, BC, & UUCC.

As I said, Condorcet Loser & Majority Loser are more about embarrassments that a method can avoid, rather than about how well it will do under typical conditions. They're met by SD & SSD, but their failure by PC and their compliance by IRV doesn't seem important.

The value of meeting the Mutual Majority Criterion is questionable. Just as in Plurality one can't always be sure of one's favorite has a majority, an IRV voter can't always know what candidates are in that criterion's favored set. When voters don't know that, they might feel compelled to compromise for a candidate outside the set, and that candidate might win. IRV has sometimes been praised for Mutual Majority. Approval & Plurality would meet Mutual Majority if it were applied to them.

IRV fails Monotonicity. That's best shown with a little story based on an example written by Professor Steven Brams:

Newspaper article the morning after a Presidential election by IRV:

"President Moe failed in his re-election bid. The only way that yesterday's rankings differ from the one that elected Moe is that this time, some people who'd previously ranked Shemp in 1st place, this time ranked him in last place. President elect Shemp gave his acceptance speech at 1:00 a.m., Eastern Time."

That's nonsense? Yes it is, and so is any method that can do that. When a method is used that can do that, how much does any election result mean?

How could this happen? Let us say that Shemp is a populist who first time round had the support many poor whites and got the highest number of first preferences. Moe came second place in first preferences but was closely followed by Rossi who was a left wing radical. Rossi's supporters gave their second preferences to the moderate Moe because they disliked Shemp's racism. However at second election Rossi persuaded some of the poor whites that Shemp was corrupt and cared nothing for ordinary folk like them. Hence Rossi got enough first preferences to push Moe into third place. Not that that helped Rossi. Many Moe supporters voted Shemp second even though they found his racism distasteful because they regarded Rossi as a dangerous communist..

I have already defined Monotonicity in the Traditional Criteria article, but I'll state it again here to remind you:

Changing your vote so as to vote a candidate lower should never make him win if he wouldn't have otherwise won. Changing your vote so as to vote a candidate higher should never make him lose if he wouldn't have otherwise lost.

Olympic 0-10:

Strategically equivalent to Approval.

Copeland:

Copeland fails the criteria in the Strategy Criteria article.

Like any pairwise-count method, Copeland meets the Condorcet Criterion, which means it meets the Majority Criterion too. Copeland meets Smith. Smith compliance implies compliance with all the numbered academic criteria listed in this article. Some authors make much of that, but those criteria are weak criteria met by lots of methods. SD & SSD have Copeland completely dominated in regards to criteria.

Dodgson:

Dodgson, being a pairwise-count method, meets the Condorcet Criterion, by itself a questionable accomplishment. It fails Smith. I haven't checked it out on the other numbered academic criteria.

Dodgson fails the criteria listed in the Strategy Criteria article.

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Let me define a criterion just for Copeland & Dodgson. First a preliminary definition:

A "clone set" is a set of candidates such that, for any voter, and for any candidate, X, outside the set, if that voter prefers any candidate in the set to X, then he prefers all the candidates in the set to X. And if he prefers X to any candidate in the set, then he prefers X to all the candidates in the set.

Weak Independence from Clones Criterion (WCC):

Adding a new candidate to the election, and to a clone set should never cause the winner to come from that clone set when it otherwise wouldn't have, unless that new candidate wins.

Say we're going to hold a vote on which movie to go to. Say there are Westerns, adventure movies, and disaster movies. Say those are clone sets, so everyone, for example, ranks all the Westerns the same with respect to all the Adventure movies. Say a majority prefer Westerns to Adventure movies, and a majority prefer adventure movies to disaster movies, and a majority prefer disaster movies to Westerns. Now say there are lots of Westerns, and very few adventure movies. This will give the election to a disaster movie. A disaster movie wins not because of anything about how many people prefer the various genres, but only because there are lots of Westerns and very few adventure movies. What kind of a way is that to choose a movie genre. Or a political party?? That's nonsense. Copeland's method is nonsense.

It's been pointed out that Copleand elections will be heavily influenced by which party can afford to run the most candidates. Copeland elections will be spirally escalations of candidate proliferation by the parties.